Subgroups — #34288841

Order 20224.

810 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23766009	C₂	1	2	yes	maximal
2	#23763564	C₂	1	1	yes	maximal, characteristic, complemented
3	#23763560	C₂	1	1	yes	maximal, characteristic, complemented
4	#23763534	C₂	1	2	yes	maximal, complemented
5	#23763433	C₂	1	2	yes	maximal
6	#23763429	C₂	1	2	yes	maximal
7	#23763297	C₂	1	2	yes	maximal, complemented
8	C₄×#14579095	C₂	1	2	yes	maximal, complemented
9	C₂×#405×C₇₉	C₂	1	1	yes	maximal, characteristic
10	C₂×#14579095	C₂²	1	2	yes	—
11	C₂×#14579095	C₂²	1	2	yes	—
12	C₂×#14579095	C₂²	1	2	yes	—
13	C₂×#14579093	C₂²	1	2	yes	—
14	#19098456	—	2	4	no	—
15	#19098449	—	2	4	no	—
16	#19098433	C₂²	1	2	yes	—
17	#19098429	C₂²	1	2	yes	—
18	#19098423	C₂²	1	4	yes	—
19	C₄×C₄⋊Dic₇₉	C₂²	1	4	yes	—
20	#19098326	C₂²	1	4	yes	—
21	#19098326	C₂²	1	4	yes	—
22	C₂²×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
23	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
24	C₂×#117×C₇₉	C₂²	1	1	yes	characteristic
25	#405×C₇₉	C₄	1	8	yes	complemented
26	C₂×#97×C₇₉	C₂²	1	1	yes	characteristic
27	C₂×#97×C₇₉	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊Dic₇₉	—	2	4	no	—
31	C₂×C₄⋊Dic₇₉	C₂³	1	4	yes	—
32	C₂×C₄⋊Dic₇₉	C₂³	1	4	yes	—
33	C₂×C₂²⋊Dic₇₉	—	2	4	no	—
34	C₂×C₄×Dic₇₉	—	2	4	no	—
35	#14579095	—	2	4	no	—
36	#14579095	—	2	4	no	—
37	#14579095	—	2	4	no	—
38	#14579093	—	2	4	no	—
39	C₇₉⋊#95	—	2	4	no	—
40	C₇₉⋊#95	—	2	4	no	—
41	C₇₉⋊#95	—	2	4	no	—
42	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
43	C₂×M₁₆×C₇₉	—	2	2	no	—
44	C₂×M₁₆×C₇₉	—	2	2	no	—
45	C₂×M₁₆×C₇₉	C₂×C₄	1	4	yes	—
46	C₂²×C₈×C₇₉	—	2	2	no	—
47	C₂²×C₈×C₇₉	—	2	2	no	—
48	C₂²×C₈×C₇₉	C₂×C₄	1	4	yes	—
49	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
53	#117×C₇₉	C₂×C₄	1	4	yes	—
54	#117×C₇₉	Q₈	1	4	yes	—
55	C₂×C₄⋊C₄×C₇₉	D₄	1	2	yes	—
56	C₂×C₂²⋊C₄×C₇₉	D₄	1	2	yes	—
57	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
59	#97×C₇₉	C₂×C₄	1	4	yes	—
60	#97×C₇₉	C₂×C₄	1	4	yes	—
61	C₄×C₈×C₇₉	C₂×C₄	1	4	yes	—
62	C₄×C₈×C₇₉	C₂×C₄	1	4	yes	—
63	C₂²×Dic₇₉	—	2	4	no	—
64	C₂²×Dic₇₉	—	2	4	no	—
65	C₄⋊Dic₇₉	—	2	4	no	—
66	C₄⋊Dic₇₉	—	2	4	no	—
67	C₄⋊Dic₇₉	—	2	4	no	—
68	C₄⋊Dic₇₉	—	2	4	no	—
69	C₄⋊Dic₇₉	—	8	4	no	—
70	C₂²⋊Dic₇₉	—	8	4	no	—
71	C₄×Dic₇₉	—	8	4	no	—
72	C₂⁴×C₇₉	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₇₉	—	2	1	no	—
74	C₂²×C₄×C₇₉	—	2	1	no	—
75	C₂²×C₄×C₇₉	—	2	1	no	—
76	C₂²×C₄×C₇₉	—	2	1	no	—
77	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₇₉	C₂×Q₈	1	2	yes	—
82	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
83	M₁₆×C₇₉	—	2	8	no	—
84	C₂×C₈×C₇₉	—	2	8	no	—
85	C₂×C₈×C₇₉	—	2	1	no	—
86	C₂×C₈×C₇₉	—	2	1	no	—
87	C₂×C₈×C₇₉	—	2	1	no	—
88	C₂×C₈×C₇₉	—	2	1	no	—
89	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
92	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
93	C₂×C₈×C₇₉	#41	1	2	yes	—
94	C₂×C₈×C₇₉	#41	1	2	yes	—
95	C₂×C₈×C₇₉	#41	1	2	yes	—
96	C₂×C₈×C₇₉	#41	1	2	yes	—
97	C₄⋊C₄×C₇₉	C₄⋊C₄	1	8	yes	—
98	C₂²⋊C₄×C₇₉	C₄⋊C₄	1	8	yes	—
99	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
100	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
101	C₄²×C₇₉	C₂×Q₈	1	2	yes	—
102	C₄²×C₇₉	C₂×Q₈	1	2	yes	—
103	C₂³×C₇₉	—	2	1	no	—
104	C₂³×C₇₉	—	2	1	no	—
105	C₂³×C₇₉	—	2	2	no	—
106	C₂³×C₇₉	—	2	2	no	—
107	C₂³×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
108	C₂³×C₇₉	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₇₉	—	2	1	no	—
110	C₂×C₄×C₇₉	—	2	1	no	—
111	C₂×C₄×C₇₉	—	2	2	no	—
112	C₂×C₄×C₇₉	—	2	2	no	—
113	C₂×C₄×C₇₉	—	2	2	no	—
114	C₂×C₄×C₇₉	—	2	2	no	—
115	C₂×C₄×C₇₉	—	2	2	no	—
116	C₂×C₄×C₇₉	—	2	2	no	—
117	C₂×C₄×C₇₉	—	2	2	no	—
118	C₂×C₄×C₇₉	—	2	2	no	—
119	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	4	yes	—
120	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	4	yes	—
121	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
122	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
124	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
125	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
126	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
127	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
128	C₂×Dic₇₉	—	4	4	no	—
129	C₂×Dic₇₉	—	4	4	no	—
130	C₂×Dic₇₉	—	4	4	no	—
131	C₂×Dic₇₉	—	4	4	no	—
132	C₂×Dic₇₉	—	4	4	no	—
133	C₂×Dic₇₉	—	4	4	no	—
134	C₈×C₇₉	—	2	2	no	—
135	C₈×C₇₉	—	2	2	no	—
136	C₈×C₇₉	#117	1	4	yes	—
137	C₈×C₇₉	#117	1	4	yes	—
138	C₂²×C₇₉	—	2	4	no	—
139	C₂²×C₇₉	—	2	2	no	—
140	C₂²×C₇₉	—	4	2	no	—
141	C₂²×C₇₉	—	4	1	no	—
142	C₂²×C₇₉	—	4	1	no	—
143	C₂²×C₇₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
144	C₂²×C₇₉	#532	1	1	yes	characteristic
145	C₂²×C₇₉	#532	1	1	yes	characteristic
146	C₂²×C₇₉	#576	1	2	yes	—
147	C₂²×C₇₉	#578	1	2	yes	—
148	C₄×C₇₉	—	2	4	no	—
149	C₄×C₇₉	—	2	4	no	—
150	C₄×C₇₉	—	2	2	no	—
151	C₄×C₇₉	C₂×#117	1	1	yes	characteristic
152	C₄×C₇₉	C₂×#117	1	1	yes	characteristic
153	C₄×C₇₉	#533	1	1	yes	characteristic
154	C₄×C₇₉	#533	1	1	yes	characteristic
155	Dic₇₉	—	8	4	no	—
156	Dic₇₉	—	8	4	no	—
157	#20821	—	79	1	no	maximal, Sylow
158	C₂×C₇₉	—	4	2	no	—
159	C₂×C₇₉	#2285	1	1	yes	characteristic
160	C₂×C₇₉	#2913	1	2	yes	—
161	C₂×C₇₉	#3175	1	2	yes	—
162	C₂×C₇₉	#3176	1	1	yes	characteristic
163	C₂×C₇₉	#3177	1	1	yes	characteristic
164	C₂×#405	D₇₉	1	1	yes	characteristic
165	#2314	—	79	2	no	—
166	#1846	—	79	1	no	—
167	#1845	—	79	1	no	—
168	#1836	—	79	2	no	—
169	#1809	—	79	2	no	—
170	#1808	—	79	2	no	—
171	#1776	—	79	2	no	—
172	C₄×#107	—	79	2	no	—
173	C₇₉	#20821	1	1	yes	characteristic, complemented, Sylow
174	C₂²×M₁₆	C₂×D₇₉	1	1	yes	characteristic
175	C₂³×C₈	C₂×D₇₉	1	1	yes	characteristic
176	C₂×#117	C₂×D₇₉	1	1	yes	characteristic
177	C₂×#107	—	79	2	no	—
178	C₂×#107	—	79	2	no	—
179	C₂×#107	—	79	2	no	—
180	C₂×#106	—	79	2	no	—
181	#405	Dic₇₉	1	8	yes	complemented
182	C₂×#97	C₂×D₇₉	1	1	yes	characteristic
183	C₂×#97	C₂×D₇₉	1	1	yes	characteristic
184	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
185	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
186	#388	—	158	4	no	—
187	#387	—	158	4	no	—
188	#384	—	79	2	no	—
189	#382	—	79	2	no	—
190	#380	—	79	4	no	—
191	C₄×C₄⋊C₄	—	79	4	no	—
192	#340	—	79	4	no	—
193	#340	—	79	4	no	—
194	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
195	C₂×M₁₆	—	2	2	no	—
196	C₂×M₁₆	—	2	2	no	—
197	C₂×M₁₆	C₂×Dic₇₉	1	4	yes	—
198	C₂²×C₈	—	2	2	no	—
199	C₂²×C₈	—	2	2	no	—
200	C₂²×C₈	C₂×Dic₇₉	1	4	yes	—
201	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
202	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
203	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
204	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
205	#117	Dic₁₅₈	1	4	yes	—
206	#117	C₂×Dic₇₉	1	4	yes	—
207	C₂×C₄⋊C₄	—	79	4	no	—
208	C₂×C₄⋊C₄	—	79	4	no	—
209	C₂×C₄⋊C₄	—	158	4	no	—
210	C₂×C₄⋊C₄	D₃₁₆	1	2	yes	—
211	C₂×C₂²⋊C₄	—	158	4	no	—
212	C₂×C₂²⋊C₄	D₃₁₆	1	2	yes	—
213	C₂×C₄²	—	158	4	no	—
214	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
215	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
216	#107	—	158	4	no	—
217	#107	—	158	4	no	—
218	#107	—	158	4	no	—
219	#106	—	158	4	no	—
220	#97	C₂×Dic₇₉	1	4	yes	—
221	#97	C₂×Dic₇₉	1	4	yes	—
222	C₄×C₈	C₂×Dic₇₉	1	4	yes	—
223	C₄×C₈	C₂×Dic₇₉	1	4	yes	—
224	#95	—	158	4	no	—
225	#95	—	158	4	no	—
226	#95	—	158	4	no	—
227	C₂⁴	C₂×D₃₁₆	1	1	yes	characteristic
228	C₂²×C₄	—	2	1	no	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	2	1	no	—
232	C₂²×C₄	—	158	4	no	—
233	C₂²×C₄	—	158	4	no	—
234	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
235	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
236	C₂²×C₄	C₂×D₃₁₆	1	2	yes	—
237	C₂²×C₄	C₂×D₃₁₆	1	2	yes	—
238	C₂²×C₄	C₂×Dic₁₅₈	1	2	yes	—
239	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
240	M₁₆	—	2	8	no	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	2	1	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	2	8	no	—
246	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
247	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
248	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
249	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
250	C₂×C₈	#11929624	1	2	yes	—
251	C₂×C₈	#11929624	1	2	yes	—
252	C₂×C₈	#11929624	1	2	yes	—
253	C₂×C₈	#11929624	1	2	yes	—
254	C₄⋊C₄	—	632	4	no	—
255	C₄⋊C₄	—	158	4	no	—
256	C₄⋊C₄	—	158	4	no	—
257	C₄⋊C₄	—	158	4	no	—
258	C₄⋊C₄	—	158	4	no	—
259	C₄⋊C₄	C₄⋊Dic₇₉	1	8	yes	—
260	C₂²⋊C₄	—	632	4	no	—
261	C₂²⋊C₄	C₄⋊Dic₇₉	1	8	yes	—
262	C₄²	—	632	4	no	—
263	C₄²	C₂²×Dic₇₉	1	2	yes	—
264	C₄²	C₂²×Dic₇₉	1	2	yes	—
265	C₄²	C₂×Dic₁₅₈	1	2	yes	—
266	C₄²	C₂×Dic₁₅₈	1	2	yes	—
267	C₂³	—	2	2	no	—
268	C₂³	—	2	1	no	—
269	C₂³	—	2	1	no	—
270	C₂³	—	2	2	no	—
271	C₂³	C₂×C₄⋊Dic₇₉	1	2	yes	—
272	C₂³	C₂²×D₃₁₆	1	1	yes	characteristic
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	316	4	no	—
284	C₂×C₄	—	316	4	no	—
285	C₂×C₄	—	316	4	no	—
286	C₂×C₄	—	316	4	no	—
287	C₂×C₄	—	316	4	no	—
288	C₂×C₄	—	316	4	no	—
289	C₂×C₄	C₂×C₄⋊Dic₇₉	1	4	yes	—
290	C₂×C₄	C₂×C₄⋊Dic₇₉	1	4	yes	—
291	C₂×C₄	C₂×C₄⋊Dic₇₉	1	2	yes	—
292	C₂×C₄	C₂³×Dic₇₉	1	1	yes	characteristic
293	C₂×C₄	C₂²×Dic₁₅₈	1	1	yes	characteristic
294	C₂×C₄	C₂×#11929624	1	1	yes	characteristic
295	C₂×C₄	C₂×#11929624	1	1	yes	characteristic
296	C₂×C₄	C₂×#11929624	1	1	yes	characteristic
297	C₂×C₄	C₂×#11929624	1	1	yes	characteristic
298	C₈	—	2	2	no	—
299	C₈	—	2	2	no	—
300	C₈	#14579120	1	4	yes	—
301	C₈	#14579120	1	4	yes	—
302	C₂²	—	2	2	no	—
303	C₂²	—	2	4	no	—
304	C₂²	—	4	2	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	4	1	no	—
307	C₂²	C₂²×C₄⋊Dic₇₉	1	1	yes	characteristic
308	C₂²	#19099113	1	1	yes	characteristic
309	C₂²	#19099113	1	1	yes	characteristic
310	C₂²	#19099368	1	2	yes	—
311	C₂²	#19099380	1	2	yes	—
312	C₄	—	2	2	no	—
313	C₄	—	2	4	no	—
314	C₄	—	2	4	no	—
315	C₄	—	632	4	no	—
316	C₄	—	632	4	no	—
317	C₄	C₂×#14579120	1	1	yes	characteristic
318	C₄	C₂×#14579120	1	1	yes	characteristic
319	C₄	#19099120	1	1	yes	characteristic
320	C₄	#19099120	1	1	yes	characteristic
321	C₂	—	4	2	no	—
322	C₂	#23765807	1	1	yes	characteristic
323	C₂	#23772000	1	2	yes	—
324	C₂	#23774039	1	2	yes	—
325	C₂	#23774050	1	1	yes	characteristic
326	C₂	#23774059	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₃₁₆
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₂×#405×C₇₉	C₂
socle	C₂³×C₇₉	C₂²×D₄
prime radical	C₂⁴×C₇₉	C₂×D₄
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	C₂×#405	D₇₉
hypocenter	C₇₉	#20821
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. #23763564	C₂	semidirect
3. #23763560	C₂	semidirect
4. #23763534	C₂	semidirect
7. #23763297	C₂	semidirect
8. C₄×#14579095	C₂	semidirect
25. #405×C₇₉	C₄	semidirect
173. C₇₉	#20821	semidirect
181. #405	Dic₇₉	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#23766009	2
#23763564	2
#23763560	2
#23763534	2
#23763433	2
#23763429	2
#23763297	2
C₄×#14579095	2
C₂×#405×C₇₉	2
#20821	79